## Automorphisms of the lattice of Π 0 1 classes: perfect thin classes and anc degrees

Venue: | Trans. Amer. Math. Soc |

Citations: | 18 - 6 self |

### BibTeX

@ARTICLE{Cholak_automorphismsof,

author = {Peter Cholak and Richard Coles and Rod Downey and Eberhard Herrmann},

title = {Automorphisms of the lattice of Π 0 1 classes: perfect thin classes and anc degrees},

journal = {Trans. Amer. Math. Soc},

year = {},

volume = {353},

pages = {4899--4924}

}

### OpenURL

### Abstract

Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin’s work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare’s work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of Π0 1 classes) forms an orbit in the lattice of Π01 classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of Π0 1 classes. We remark that the automorphism result is proven via a ∆0 3 automorphism, and demonstrate that this complexity is necessary. 1.

### Citations

896 |
Jr.; Theory of Recursive Functions and Effective Computability
- Rogers
- 1987
(Show Context)
Citation Context ...e Boolean algebra Be is isomorphic to the computable Boolean algebra Bi are Σ1 1-complete. Since we know of no explicit proof of this in the literature, here is a proof. It is well-known (e.g. Rogers =-=[25]-=-) that the collection of paths through Kleene’s O is Σ 1 1 complete. For each computable ordinal β, construct a canonical computable Boolean algebra as the interval algebra Intal(ω β ). These algebras... |

138 |
01 classes and degrees of theories
- Soare
- 1972
(Show Context)
Citation Context ...if I is a computable ideal in a computable ring R then the collection of prime ideals containing I forms a Π 0 1 class. Π 0 1-classes have been studied for many years (for example, Jockusch and Soare =-=[18]-=-, Kreisel [19]) and are particularly associated with models of Peano arithmetic (PA) and proof theoretical notions such as WKL0. There are many ways of viewing Π0 1 classes and many connections of com... |

123 | Recursively enumerable sets of positive integers and their decision problems - Post - 1944 |

56 |
Classes of Recursively Enumerable Sets and Degrees of Unsolvability
- Martin
- 1966
(Show Context)
Citation Context ...e lattice of computably enumerable sets which guaranteed Turing incompleteness. In the deep paper [28], Soare demonstrated that this was impossible since all maximal sets were automorphic, and Martin =-=[21]-=- had earlier proved that the degrees containing maximal sets (indeed hyperhypersimple sets) were precisely the collection of all high degrees. In the present paper we initiate a program similar to the... |

33 | Stob; Automorphisms of the Lattice of Recursively Enumerable Sets
- Downey, M
- 1992
(Show Context)
Citation Context ...27]) conjecture holds: Open Question. Is {〈e, i〉 : Pe automorphic to Pi} Σ 1 1 complete? The analogous result for the lattice of computably enumerable sets was proven by Cholak, Downey and Harrington =-=[6]-=-. 5. Perfect Thin Classes We now turn to perfect classes, our main concern. Recall that for a topological space X, the set X d of derived points of X is the set of x such that x ∈ cl(X −{x}). If x ∈ X... |

26 | Array nonrecursive degrees and genericity
- Downey, Jockusch, et al.
- 1996
(Show Context)
Citation Context ... and B2 have the property that A1 ⊕ A2 ⊕ B1 ⊕ B2 is of anc degree. Furthermore, if a is anc then a contains a 4-tuple of this form. A number of further results on anc degrees can be found in [12] and =-=[13]-=-. Of relevance to us here are the following results from Downey, Jockusch and Stob [12]. Theorem 7.3 (Downey, Jockusch and Stob). If M is a Martin–Pour-El theory, then M has anc degree. Theorem 7.4 (D... |

22 |
Automorphisms of the lattice of recursively enumerable sets, Part I: Maximal sets
- Soare
- 1974
(Show Context)
Citation Context ...ree notions along the lines of Post’s program. As is well known, Post sought a thinness property of the lattice of computably enumerable sets which guaranteed Turing incompleteness. In the deep paper =-=[28]-=-, Soare demonstrated that this was impossible since all maximal sets were automorphic, and Martin [21] had earlier proved that the degrees containing maximal sets (indeed hyperhypersimple sets) were p... |

20 |
Hierarchies of Boolean Algebras
- Feiner
- 1970
(Show Context)
Citation Context ...ere F ≡ H iff 〈G ∪ F 〉 = 〈G ∪ H〉. Note that ≡ is clearly c.e. The induced +, ∩ and complementation are obvious ones. For the converse direction, let B be a ∆0 2 Boolean algebra. By a result of Feiner =-=[14]-=- (see Downey [10], Corollary 3.10), we know that B is isomorphic to a c.e. presented Boolean algebra. Hence we can suppose, without loss of generality, that there is a c.e. theory F such that B ∼ = Q/... |

19 | Recursively Enumerable Sets and Degrees (Springer-Verlag - Soare - 1987 |

19 |
Π 0 1 classes in mathematics
- Cenzer, Remmel
- 1998
(Show Context)
Citation Context ...e important connections more explicit in the next section.) recent extensive surveys on Π 0 1 classes and their applications, we refer the reader to Cenzer [2], Cenzer-Jockusch [4], and Cenzer-Remmel =-=[5]-=-. The collection of Π 0 1 classes form a lattice L(2 ω ). In this paper we study this lattice and its connection with the computably enumerable degrees, along the same lines as the well known Post pro... |

18 |
Axiomatixable Theories with Few Axiomatizable Extensions
- Martin, Pour-El
- 1970
(Show Context)
Citation Context ...e same way that the study of E does. The central concept of the present paper is that of a thin Π 0 1-class, which corresponds to a “maximal”, or perhaps “hyperhypersimple” theory. Martin and Pour-El =-=[22]-=- constructed a perfect c.e. theory A (that is, essentially undecidable, or, viewed as a c.e. filter in the free boolean algebra, the quotient of Q by A would be isomorphic to Q) which was maximal. Tha... |

17 | On presentations of algebraic structures - Downey - 1997 |

15 |
Note on arithmetical models for consistent formulae of the predicate calculus
- Kreisel
- 1950
(Show Context)
Citation Context ...utable ideal in a computable ring R then the collection of prime ideals containing I forms a Π 0 1 class. Π 0 1-classes have been studied for many years (for example, Jockusch and Soare [18], Kreisel =-=[19]-=-) and are particularly associated with models of Peano arithmetic (PA) and proof theoretical notions such as WKL0. There are many ways of viewing Π0 1 classes and many connections of computable mathem... |

11 | Abstract Dependence, Recursion Theory and the Lattice of Recursively Enumerable Filters - Downey - 1982 |

9 |
Π01 classes in computability theory
- Cenzer
- 1999
(Show Context)
Citation Context ... all standard, we will make some of these important connections more explicit in the next section.) For recent extensive surveys on Π 0 1 classes and their applications, we refer the reader to Cenzer =-=[2]-=-, Cenzer-Jockusch [4], and Cenzer-Remmel [5]. The collection of Π 0 1 classes forms a lattice L(2ω ). In this paper we study this lattice and its connection with the computably enumerable degrees, alo... |

9 |
Π01 classes—structure and applications, in Computability Theory and its
- Cenzer, Jockusch
(Show Context)
Citation Context ...l make some of these important connections more explicit in the next section.) For recent extensive surveys on Π 0 1 classes and their applications, we refer the reader to Cenzer [2], Cenzer-Jockusch =-=[4]-=-, and Cenzer-Remmel [5]. The collection of Π 0 1 classes forms a lattice L(2ω ). In this paper we study this lattice and its connection with the computably enumerable degrees, along the same lines as ... |

7 |
Maximal theories, Annals of Pure and
- Downey
- 1987
(Show Context)
Citation Context ...e we say C is thin if it is infinite and for all Π 0 1 subclasses C′ there is a clopen U such that C ′ = C ∩ U. What are the basic degree theoretical properties of thin classes? In his thesis, Downey =-=[9]-=- proved that not every degree contains a Martin–Pour-El theory. He showed that while all high degrees contained Martin–Pour-El theories, and some low degrees, there were initial segments not containin... |

6 | Countable thin Π01 classes
- Cenzer, Downey, et al.
- 1993
(Show Context)
Citation Context ...We remark that Open Question 6.3 in Cenzer-Jockusch [4] asks whether “T is finite” is definable in L(2ω ). For more results and background on thin Π0 1-classes, see Cenzer, Downey, Jockusch and Shore =-=[3]-=- and Downey [9]. The following says that, in a sense, thin classes are the precise analogues of hyper-hyper-simple c.e. sets. Lemma 4.3. A nonempty Π0 1-class P is thin if and only if L(2ω )(P, ↓) is ... |

6 | Effective presentability of Boolean algebras of Cantor–Bendixson rank 1
- Downey, Jockusch
- 1999
(Show Context)
Citation Context ...ove). This would require significant technology, since there are computable Boolean algebras that are not even arithmetically isomorphic. This is even true of rank 1 Boolean algebras (Downey-Jockusch =-=[11]-=-). A good test case is to consider whether, when B1 and B2 are computable copies of the Boolean algebra of finite and cofinite sets, are B1 and B2 automorphic? 3 The proof technique from Theorem 7.6 i... |

6 | Posts’s program and incomplete recursively enumerable sets - Harrington, Soare - 1991 |

5 |
Private communication
- Remmel
(Show Context)
Citation Context ...computable automorphism of Q. Moreover, we can construct two thin Π0 1-classes that are not automorphic via a ∆0 2 automorphism. This seems to present an obstacle to our program. Nevertheless, Remmel =-=[24]-=- has proved that L(2ω )has2ℵ0automorphisms. We give a proof of Remmel’s theorem in section 6. So there is some hope of an analogue of Soare’s theorem. Indeed, as we show, this is the case. Using some ... |

4 | Automorphisms of the lattice of recursively enumerable sets and hyperhypersimple sets - Herrmann - 1989 |

3 |
Array nonrecursice sets and multiple permitting arguments
- Downey, Jockusch, et al.
- 1990
(Show Context)
Citation Context ...is, each follower needs more permissions than its predecessor for followers of the same requirement. Eventually a new degree class, called the anc degrees, was introduced by Downey, Jockusch and Stob =-=[12]-=- to explain such arguments. It turns out that the anc degrees are a class containing all nonlow2 degrees, and are closed upwards. They are exactly the degrees realized by many known constructions. We ... |

3 | A maximal set which is not complete - Sacks - 1964 |

1 |
Automorphisms of substructure lattices in effective algebra, Ann. Pure and Applied Logic 25
- Guichard
- 1983
(Show Context)
Citation Context ...esults for constructing automorphisms of L(2ω ). Evidence from other structures said that the presence of additional algebraic structure can lead to quite different situations. For instance, Guichard =-=[15]-=- proved that the lattice of c.e. subspaces of an infinite dimensional fully effective vector space L(V∞) has only countably many automorphisms, each induced by a computable semi-linear transformation ... |

1 | A maximal set that is not complete - Sacks - 1964 |